# Ideal Geometric Arrangement of Microphone Array

I'm going to have to give a bit of context for this question to make sense.

I am working on a project which includes audio source localisation in 3-D space through TDoA (Time Difference of Arrival - from here called $\tau_{ij}$, between sensor $i$ and $j$) measurements. In other words I have $m$ sensors that each have a slightly different recording of the audio event. Via cross-correlation methods I can estimate the TDoA (or delay) between each pair of sensors.

With each pair of sensors, $i$ and $j$, we can point to the source position through the following optimization problem:

$$\delta_{ij} = v \tau_{ij}$$ $$h(P_{E},P_{i},P_{j}) = \|P_i - P_E\|_2 - \|P_j - P_E\|_2$$ $$P_E^* = {arg\,min}_{P_E}\sum\limits_{i \neq j}\left(\delta_{ij}-h(P_{E},P_{i},P_{j})\right)^2$$ where $v$ is the speed of sound in the selected medium (consider it ideally constant), $P_E^* = (x_E,y_E,z_E)$ is the source position estimate (after optimization), while $P_i = (x_i,y_i,z_i)$ and $P_j = (x_j,y_j,z_j)$ are the sensor positions.

Note the first two equations. The first one is the distance between the two sensors, calculated from the TDoA estimate. The second one is a hyperbolic surface (see image below, in a 2-D example) that also represents the distance between the sensors. One way to see this is noting (suppose perfect measurements):

$$\|P_i - P_E\|_2 - \|P_j - P_E\| = v \tau_{iE} - v \tau_{jE} = v (\tau_{iE} - \tau_{jE}) = \delta_{ij}$$

In other words, each pair of sensors generate a different hyperbolic surface. We then try find the position by minimizing the function (third equation), by varying $P_E$. For sure there will always be a $P_E$ where $\delta_{ij} = h(P_E,P_i,P_j)$, but it won't necessarily nullify the same equation for different $i$ and $j$. Therefore optimization is necessary. In a simpler 2-D example, the hyperbolic curves above represent points where the TDoA between two sensors is constant.

Finally, my question is: What is the best geometric arrangement of the $m$ sensors in 3-D space, so that measurements are as precise as possible? For example, a line of sensors is probably not a good idea as the the hyperbolic curves will be parallel to each other.

• Yeah, for 3 sensors don't put them all in a line, or you won't have 2D information, and for 4 sensors don't put them all in a plane, or you won't have 3D information. "that each have a slightly different recording of the audio event" So there's a constraint that the sensors have to be nearer each other than they are to the source? You can't place them around the corners of a room so that the source is in the middle of the sensors, for instance? Because otherwise I suspect making them as far apart as possible is the best solution for resolution (but not SNR) – endolith Jul 1 '13 at 23:27
• Well technically its an underwater source localisation problem. There are no corners. :P I am trying to devise an algorithm for the movement of an array of sensors to get as close to the source as possible. – jhc Jul 1 '13 at 23:29
• so the sensors are fixed relative to each other and have to move as one unit? they can't be independent? – endolith Jul 1 '13 at 23:31
• They are not fixed. I'm just asking because I'm sure there's an ideal arrangement. I imagine it would be one that minimizes the sum of the dot products of all the sensor pairs. In other words a simplex: en.wikipedia.org/wiki/Simplex – jhc Jul 1 '13 at 23:36
• @jhc: Easiest to see in 1D, but the idea generalizes. Assume you have a linear array with microphones at positions 0, 1 meter & 2 meter, and a periodic signal with wavelength 0.50 meters coming in from the left side. You wouldn't be able to tell left&right apart because all three signals would be in phase. If you put the third microphone at sqrt(2), the 3 inputs can´t be in phase regardless of wavelength. – MSalters Jul 4 '13 at 8:03